Can I control hair thickness with weight values?
There is a way to add Alpha to the tips of the hairs, using a Hair Info Node. This will require you to use Cycles. 
I just pulled this here straight out of the Cycles Encyclopedia by Blender Guru. 
https://www.blenderguru.com/articles/cycles-input-encyclopedia





You can also use the Strand Settings to scale all hair equally.



I am checking to see whether you can feed in a texture or a vertex map to drive the render attributes, so you could paint in the width of the scale of the individual strands. Can't tell you off the cuff if that is possible, but I would find it useful, too.